Projective bundle and blow up

Consider $mathbb{P}^2$ and a point $pinmathbb{P}^2$. We can construct the blow-up of $mathbb{P}^2$ at $p$ (we denote it by $tilde{mathbb{P}}^2$) as the closed subvariety
$$tilde{mathbb{P}}^2subset mathbb{P}^2timesmathbb{P}^1$$
defined as $tilde{mathbb{P}}^2={(u,v)mid v_iu_j=u_iv_j }$
(More precisely I think the blow-up is the variety $tilde{mathbb{P}}^2$ together with the surjective morphism $tilde{mathbb{P}}^2to mathbb{P^2}$)

On the same time I can consider the map $tilde{mathbb{P}}^2tomathbb{P}^1$, which I was able to prove to be a $mathbb{P}^1$ bundle over $mathbb{P}^1$.

Question: In the handwritten notes of the lecturer there is written it can be showed that there exists an isomorphism
$$tilde{mathbb{P}}^2simeq mathbb{P}(mathcal{O}oplusmathcal{O}(1))to mathbb{P}^1,$$
and I’d like to understand

• Why this holds (mathematiclaly and geometrically spekaing, I have no idea of how to visualize or prove it)
• Does it hold in a mor general setting (that is, for $mathbb{P}^n$ or even for a smooth subvariety $Ysubset mathbb{P}^n$)?

Even a reference would be sufficient, thanks in advance. (this is not material for an exam or homework, just my curiosity)